Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(5,6,7,8)(10,14)(11,12)(15,16);(9,10)(11,14)(12,16)(13,15);(2,4)(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^7|a^2=f,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg38 conjugacy classes — view the full character table.
| maximal subgroup of | #8465, #8467, #9345, … |
| maximal normal subgroup of | #8465, #8467, #9345, … |
| maximal quotient of | #8312, #8315, #8330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 520);